3.343 \(\int \frac{\log ^3(c (a+b x)^n)}{d x+e x^2} \, dx\)

Optimal. Leaf size=238 \[ \frac{6 n^2 \log \left (c (a+b x)^n\right ) \text{PolyLog}\left (3,-\frac{e (a+b x)}{b d-a e}\right )}{d}-\frac{3 n \log ^2\left (c (a+b x)^n\right ) \text{PolyLog}\left (2,-\frac{e (a+b x)}{b d-a e}\right )}{d}-\frac{6 n^2 \text{PolyLog}\left (3,\frac{b x}{a}+1\right ) \log \left (c (a+b x)^n\right )}{d}+\frac{3 n \text{PolyLog}\left (2,\frac{b x}{a}+1\right ) \log ^2\left (c (a+b x)^n\right )}{d}-\frac{6 n^3 \text{PolyLog}\left (4,-\frac{e (a+b x)}{b d-a e}\right )}{d}+\frac{6 n^3 \text{PolyLog}\left (4,\frac{b x}{a}+1\right )}{d}-\frac{\log ^3\left (c (a+b x)^n\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{d}+\frac{\log \left (-\frac{b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d} \]

[Out]

(Log[-((b*x)/a)]*Log[c*(a + b*x)^n]^3)/d - (Log[c*(a + b*x)^n]^3*Log[(b*(d + e*x))/(b*d - a*e)])/d - (3*n*Log[
c*(a + b*x)^n]^2*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/d + (3*n*Log[c*(a + b*x)^n]^2*PolyLog[2, 1 + (b*x)/
a])/d + (6*n^2*Log[c*(a + b*x)^n]*PolyLog[3, -((e*(a + b*x))/(b*d - a*e))])/d - (6*n^2*Log[c*(a + b*x)^n]*Poly
Log[3, 1 + (b*x)/a])/d - (6*n^3*PolyLog[4, -((e*(a + b*x))/(b*d - a*e))])/d + (6*n^3*PolyLog[4, 1 + (b*x)/a])/
d

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Rubi [A]  time = 0.35135, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {1593, 2416, 2396, 2433, 2374, 2383, 6589} \[ \frac{6 n^2 \log \left (c (a+b x)^n\right ) \text{PolyLog}\left (3,-\frac{e (a+b x)}{b d-a e}\right )}{d}-\frac{3 n \log ^2\left (c (a+b x)^n\right ) \text{PolyLog}\left (2,-\frac{e (a+b x)}{b d-a e}\right )}{d}-\frac{6 n^2 \text{PolyLog}\left (3,\frac{b x}{a}+1\right ) \log \left (c (a+b x)^n\right )}{d}+\frac{3 n \text{PolyLog}\left (2,\frac{b x}{a}+1\right ) \log ^2\left (c (a+b x)^n\right )}{d}-\frac{6 n^3 \text{PolyLog}\left (4,-\frac{e (a+b x)}{b d-a e}\right )}{d}+\frac{6 n^3 \text{PolyLog}\left (4,\frac{b x}{a}+1\right )}{d}-\frac{\log ^3\left (c (a+b x)^n\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{d}+\frac{\log \left (-\frac{b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x)^n]^3/(d*x + e*x^2),x]

[Out]

(Log[-((b*x)/a)]*Log[c*(a + b*x)^n]^3)/d - (Log[c*(a + b*x)^n]^3*Log[(b*(d + e*x))/(b*d - a*e)])/d - (3*n*Log[
c*(a + b*x)^n]^2*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/d + (3*n*Log[c*(a + b*x)^n]^2*PolyLog[2, 1 + (b*x)/
a])/d + (6*n^2*Log[c*(a + b*x)^n]*PolyLog[3, -((e*(a + b*x))/(b*d - a*e))])/d - (6*n^2*Log[c*(a + b*x)^n]*Poly
Log[3, 1 + (b*x)/a])/d - (6*n^3*PolyLog[4, -((e*(a + b*x))/(b*d - a*e))])/d + (6*n^3*PolyLog[4, 1 + (b*x)/a])/
d

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2396

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*
(f + g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n])^p)/g, x] - Dist[(b*e*n*p)/g, Int[(Log[(e*(f + g*x))/(e*f -
d*g)]*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*
f - d*g, 0] && IGtQ[p, 1]

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{\log ^3\left (c (a+b x)^n\right )}{d x+e x^2} \, dx &=\int \frac{\log ^3\left (c (a+b x)^n\right )}{x (d+e x)} \, dx\\ &=\int \left (\frac{\log ^3\left (c (a+b x)^n\right )}{d x}-\frac{e \log ^3\left (c (a+b x)^n\right )}{d (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{\log ^3\left (c (a+b x)^n\right )}{x} \, dx}{d}-\frac{e \int \frac{\log ^3\left (c (a+b x)^n\right )}{d+e x} \, dx}{d}\\ &=\frac{\log \left (-\frac{b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac{\log ^3\left (c (a+b x)^n\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{d}-\frac{(3 b n) \int \frac{\log \left (-\frac{b x}{a}\right ) \log ^2\left (c (a+b x)^n\right )}{a+b x} \, dx}{d}+\frac{(3 b n) \int \frac{\log ^2\left (c (a+b x)^n\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{a+b x} \, dx}{d}\\ &=\frac{\log \left (-\frac{b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac{\log ^3\left (c (a+b x)^n\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{d}-\frac{(3 n) \operatorname{Subst}\left (\int \frac{\log ^2\left (c x^n\right ) \log \left (-\frac{b \left (-\frac{a}{b}+\frac{x}{b}\right )}{a}\right )}{x} \, dx,x,a+b x\right )}{d}+\frac{(3 n) \operatorname{Subst}\left (\int \frac{\log ^2\left (c x^n\right ) \log \left (\frac{b \left (\frac{b d-a e}{b}+\frac{e x}{b}\right )}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{d}\\ &=\frac{\log \left (-\frac{b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac{\log ^3\left (c (a+b x)^n\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{d}-\frac{3 n \log ^2\left (c (a+b x)^n\right ) \text{Li}_2\left (-\frac{e (a+b x)}{b d-a e}\right )}{d}+\frac{3 n \log ^2\left (c (a+b x)^n\right ) \text{Li}_2\left (1+\frac{b x}{a}\right )}{d}-\frac{\left (6 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^n\right ) \text{Li}_2\left (\frac{x}{a}\right )}{x} \, dx,x,a+b x\right )}{d}+\frac{\left (6 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^n\right ) \text{Li}_2\left (-\frac{e x}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{d}\\ &=\frac{\log \left (-\frac{b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac{\log ^3\left (c (a+b x)^n\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{d}-\frac{3 n \log ^2\left (c (a+b x)^n\right ) \text{Li}_2\left (-\frac{e (a+b x)}{b d-a e}\right )}{d}+\frac{3 n \log ^2\left (c (a+b x)^n\right ) \text{Li}_2\left (1+\frac{b x}{a}\right )}{d}+\frac{6 n^2 \log \left (c (a+b x)^n\right ) \text{Li}_3\left (-\frac{e (a+b x)}{b d-a e}\right )}{d}-\frac{6 n^2 \log \left (c (a+b x)^n\right ) \text{Li}_3\left (1+\frac{b x}{a}\right )}{d}+\frac{\left (6 n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{x}{a}\right )}{x} \, dx,x,a+b x\right )}{d}-\frac{\left (6 n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{e x}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{d}\\ &=\frac{\log \left (-\frac{b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac{\log ^3\left (c (a+b x)^n\right ) \log \left (\frac{b (d+e x)}{b d-a e}\right )}{d}-\frac{3 n \log ^2\left (c (a+b x)^n\right ) \text{Li}_2\left (-\frac{e (a+b x)}{b d-a e}\right )}{d}+\frac{3 n \log ^2\left (c (a+b x)^n\right ) \text{Li}_2\left (1+\frac{b x}{a}\right )}{d}+\frac{6 n^2 \log \left (c (a+b x)^n\right ) \text{Li}_3\left (-\frac{e (a+b x)}{b d-a e}\right )}{d}-\frac{6 n^2 \log \left (c (a+b x)^n\right ) \text{Li}_3\left (1+\frac{b x}{a}\right )}{d}-\frac{6 n^3 \text{Li}_4\left (-\frac{e (a+b x)}{b d-a e}\right )}{d}+\frac{6 n^3 \text{Li}_4\left (1+\frac{b x}{a}\right )}{d}\\ \end{align*}

Mathematica [B]  time = 0.210555, size = 494, normalized size = 2.08 \[ \frac{-3 n^2 \left (n \log (a+b x)-\log \left (c (a+b x)^n\right )\right ) \left (2 \text{PolyLog}\left (3,\frac{e (a+b x)}{a e-b d}\right )-2 \log (a+b x) \text{PolyLog}\left (2,\frac{e (a+b x)}{a e-b d}\right )-2 \text{PolyLog}\left (3,\frac{b x}{a}+1\right )+2 \log (a+b x) \text{PolyLog}\left (2,\frac{b x}{a}+1\right )-\log ^2(a+b x) \log \left (\frac{b (d+e x)}{b d-a e}\right )+\log \left (-\frac{b x}{a}\right ) \log ^2(a+b x)\right )+3 n \left (\log \left (c (a+b x)^n\right )-n \log (a+b x)\right )^2 \left (-\text{PolyLog}\left (2,\frac{e (a+b x)}{a e-b d}\right )-\text{PolyLog}\left (2,-\frac{b x}{a}\right )-\log (a+b x) \log \left (\frac{b (d+e x)}{b d-a e}\right )+\log (x) \left (\log (a+b x)-\log \left (\frac{b x}{a}+1\right )\right )\right )+n^3 \left (-6 \text{PolyLog}\left (4,\frac{e (a+b x)}{a e-b d}\right )-3 \log ^2(a+b x) \text{PolyLog}\left (2,\frac{e (a+b x)}{a e-b d}\right )+6 \log (a+b x) \text{PolyLog}\left (3,\frac{e (a+b x)}{a e-b d}\right )+6 \text{PolyLog}\left (4,\frac{b x}{a}+1\right )+3 \log ^2(a+b x) \text{PolyLog}\left (2,\frac{b x}{a}+1\right )-6 \log (a+b x) \text{PolyLog}\left (3,\frac{b x}{a}+1\right )-\log ^3(a+b x) \log \left (\frac{b (d+e x)}{b d-a e}\right )+\log \left (-\frac{b x}{a}\right ) \log ^3(a+b x)\right )+\log (d+e x) \left (n \log (a+b x)-\log \left (c (a+b x)^n\right )\right )^3-\log (x) \left (n \log (a+b x)-\log \left (c (a+b x)^n\right )\right )^3}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x)^n]^3/(d*x + e*x^2),x]

[Out]

(-(Log[x]*(n*Log[a + b*x] - Log[c*(a + b*x)^n])^3) + (n*Log[a + b*x] - Log[c*(a + b*x)^n])^3*Log[d + e*x] + 3*
n*(-(n*Log[a + b*x]) + Log[c*(a + b*x)^n])^2*(Log[x]*(Log[a + b*x] - Log[1 + (b*x)/a]) - Log[a + b*x]*Log[(b*(
d + e*x))/(b*d - a*e)] - PolyLog[2, -((b*x)/a)] - PolyLog[2, (e*(a + b*x))/(-(b*d) + a*e)]) - 3*n^2*(n*Log[a +
b*x] - Log[c*(a + b*x)^n])*(Log[-((b*x)/a)]*Log[a + b*x]^2 - Log[a + b*x]^2*Log[(b*(d + e*x))/(b*d - a*e)] -
2*Log[a + b*x]*PolyLog[2, (e*(a + b*x))/(-(b*d) + a*e)] + 2*Log[a + b*x]*PolyLog[2, 1 + (b*x)/a] + 2*PolyLog[3
, (e*(a + b*x))/(-(b*d) + a*e)] - 2*PolyLog[3, 1 + (b*x)/a]) + n^3*(Log[-((b*x)/a)]*Log[a + b*x]^3 - Log[a + b
*x]^3*Log[(b*(d + e*x))/(b*d - a*e)] - 3*Log[a + b*x]^2*PolyLog[2, (e*(a + b*x))/(-(b*d) + a*e)] + 3*Log[a + b
*x]^2*PolyLog[2, 1 + (b*x)/a] + 6*Log[a + b*x]*PolyLog[3, (e*(a + b*x))/(-(b*d) + a*e)] - 6*Log[a + b*x]*PolyL
og[3, 1 + (b*x)/a] - 6*PolyLog[4, (e*(a + b*x))/(-(b*d) + a*e)] + 6*PolyLog[4, 1 + (b*x)/a]))/d

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Maple [C]  time = 1.178, size = 12205, normalized size = 51.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x+a)^n)^3/(e*x^2+d*x),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (b x + a\right )}^{n} c\right )^{3}}{e x^{2} + d x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x+a)^n)^3/(e*x^2+d*x),x, algorithm="maxima")

[Out]

integrate(log((b*x + a)^n*c)^3/(e*x^2 + d*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (b x + a\right )}^{n} c\right )^{3}}{e x^{2} + d x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x+a)^n)^3/(e*x^2+d*x),x, algorithm="fricas")

[Out]

integral(log((b*x + a)^n*c)^3/(e*x^2 + d*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (c \left (a + b x\right )^{n} \right )}^{3}}{x \left (d + e x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(b*x+a)**n)**3/(e*x**2+d*x),x)

[Out]

Integral(log(c*(a + b*x)**n)**3/(x*(d + e*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (b x + a\right )}^{n} c\right )^{3}}{e x^{2} + d x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x+a)^n)^3/(e*x^2+d*x),x, algorithm="giac")

[Out]

integrate(log((b*x + a)^n*c)^3/(e*x^2 + d*x), x)